Long-range correlations and generic scale invariance in classical fluids
and disordered electron systems
arXiv:cond-mat/9602144v2 [cond-mat.stat-mech] 2 Jun 1997
T.R.Kirkpatrick
Institute for Physical Science and Technology, and Department of Physics
University of Maryland,
College Park, MD 20742
D.Belitz
Department of Physics and Materials Science Institute
University of Oregon,
Eugene, OR 97403
(August 13, 2013)
Long-ranged, or power-law, behavior of correlation functions in both space and time is discussed
for classical systems and for quantum systems at finite temperature, and is compared with the
corresponding behavior in quantum systems at zero temperature. The origin of the long-ranged
correlations is explained in terms of soft modes. In general, correlations at zero temperature are
of longer range than their finite temperature or classical counterparts. This phenomenon is due to
additional soft modes that exist at zero temperature.
above phenomena in classical fluids and Lorentz gases.
In Sec. III we discuss the analogous effects in a quantum
system, namely disordered electrons. In general, QGSI
is characterized by correlations of longer range in space
and/or time than those characteristic of the classical GSI.
Reasons for this are discussed. In Sec. IV we conclude
with a summary and a few remarks.
I. INTRODUCTION
Homogeneous functions, or power laws, of space and
time do not contain any intrinsic length or time scales,
in contrast to, e.g., exponentials. This so-called scale
invariance is well known to occur at critical points, where
the critical modes become soft, which leads to powerlaw correlation functions.1 Critical points are exceptional
points in the phase diagrams of materials, and reaching
them requires the fine tuning of parameter values. What
is less well appreciated is the fact that many systems
display what is now known as generic scale invariance
(GSI), that is, power-law correlation functions in large
regions of parameter space, with no fine tuning required
at all to see them. GSI is due to soft modes that are
not related to critical phenomena, but rather are due
to conservation laws, or are otherwise inherent to the
system. In recent years there has been a lot of attention
paid to GSI, and the phenomenon has been discussed in
a wide variety of systems, ranging from classical fluids
and liquid crystals to disordered electrons and sandpiles.
For recent reviews see Refs. 2.
In this paper we compare and contrast GSI and related phenomena in classical fluids and Lorentz gases
with what we will call quantum GSI (QGSI), a similar but in general stronger effect that occurs in quantum systems at zero temperature. We will discuss four
closely related topics: (1) The non-existence of virial
expansions for the transport coefficients, (2) long-time
tails, i.e. power-law decays of equilibrium time correlation functions that determine the transport coefficients,
(3) long-range, or power-law, spatial correlations in nonequilibrium steady states, and (4) power-law spatial correlations in quantum mechanical systems in and out of
equilibrium.
In Sec. II of this paper we review the occurrence of the
II. REVIEW OF CLASSICAL SYSTEMS
A. Density expansion of the transport coefficients
The first indications for long-ranged dynamical correlations in classical fluids appeared in the 1960’s, when
problems were encountered in attempts to theoretically
establish the density dependence of the transport coefficients of moderately dense gases. Up to that time it
had been assumed that, in analogy with the virial expansion for the thermodynamic quantities3 such as, e.g.,
the pressure, an analytic density expansion existed for an
arbitrary transport coefficient, η, of the form,
η/ηB = 1 + a1,η n∗ + a2,η (n∗ )2 + O (n∗ )3
. (2.1)
Here ηB is the Boltzmann value for η, which is exact in
the limit of vanishing fluid density, n → 0. n∗ = nσ d is
the reduced density, with σ a molecular length scale and
d the spatial dimensionality of the system. The numbers
al,η are the coefficients in the virial expansion of η.
A density expansion of the form of Eq. (2.1) was predicted by Bogoliubov’s theory,4 an extension and generalization of Boltzmann’s kinetic theory to higher densities. This theory was generally accepted at the time,
and it gave the coefficients al,η in terms of formally exact integrals. It came as a big surprise when a detailed
1
analysis of these integrals showed that al,η diverges for
l ≥ 1 in two-dimensional (2-d) systems, and for l ≥ 2
in 3-d systems.5 Shortly after this discovery, exact calculations for a model fluid, the classical Lorentz gas, confirmed these results.6 A Lorentz gas consists of noninteracting particles that move in an array of randomly positioned, static hard disk (d = 2) or hard sphere (d = 3)
scatterers.7 The moving particles can be either classical
or quantum in nature. The only relevant transport coefficient is the diffusion coefficient, or, equivalently, the particle mobility or conductivity. The important point was
that the exact explicit calculations for classical Lorentz
gases showed that the coefficients al,D in the virial expansion, Eq. (2.1), for the diffusion coefficient are indeed
divergent for l ≥ l∗ , and that the value of l∗ displays the
dimensionality dependence suggested by the estimates for
real gases. For later reference, we mention that the quantum version of the Lorentz gas is essentially the standard
Edwards model for systems of disordered, noninteracting
electrons.8
These divergencies in the virial coefficients are a clear
indication of the presence of long-ranged dynamical correlations in both Lorentz gases and real gases. They have
the following origin: In making a virial expansion, one
assumes that the l-th term in Eq. (2.1) is determined
by the dynamics of a cluster of l + 2 particles moving
in the infinite system. Because the cluster is considered
in isolation, the l + 2 particles can travel over arbitrarily large distances between collisions. However, this can
occur neither in a real gas nor in a Lorentz gas, since
the presence of the other particles that are not members
of the cluster under consideration means that a particle
cannot travel farther than a distance on the order of a
mean-free path before it collides with another particle.
This is a collective many-particle effect that was missed
in Bogoliubov’s cluster expansion. Its existence precludes
a virial expansion for transport coefficients. Mathematically, this physical effect leads to a nonanalytic density
dependence of the transport coefficients, i.e. the virial coefficients are not simply numbers, but nonanalytic functions of the density. The leading nonanalyticity turns
out to be logarithmic,9 in agreement with the logarithmic divergence found in the early work. Equation (2.1)
then takes the form, in d = 3,
transport coefficients decay exponentially with time for
long times. This belief was based again on Bogoliubov’s
kinetic theory, which predicted such an exponential decay. Actually, the separation of time scales that is present
only with exponentially decaying time correlations was
an important presumption in that theory. This appeared
to be very plausible since the Boltzmann equation, which
becomes exact in the limit of dilute gases, also yields an
exponential decay of time correlations, which seemed to
guarantee an exponential decay at least for dilute gases.
It was therefore a further completely unexpected development when Alder and Wainwright,11 in a computer
study of self-diffusion in systems of hard disks and hard
spheres, discovered that the equilibrium velocity autocorrelation function, hv(0) · v(t)ieq , whose time integral
determines the diffusion coefficient, D, via
Z
1 ∞
dt hv(0) · v(t)ieq ,
(2.3)
D=
d 0
decays only algebraically with time, namely,
hv(0) · v(t)ieq ≈ c (t0 /t)d/2
(for
t >> t0 ) ,
(2.4)
where t0 is the mean-free time between collisions. Here
< . . . >eq denotes an equilibrium thermal average. This
slow decay of correlation functions is known as a longtime tail (LTT). The constant c is positive, which implies
that the LTT contribution to the autocorrelation function increases the diffusion rate compared to the Boltzmann result. Note that Eqs. (2.3) and (2.4) imply that
the diffusion coefficient D does not exist for d ≤ 2, and
that for low frequencies, the frequency dependent diffusion coefficient, D(ω), behaves as,
D(ω)/D0 = 1 − c′ (iω)(d−2)/2 + . . .
D(ω)/D0 = 1 − c′′ log(iω) + . . .
(d > 2) , (2.5a)
(d = 2) ,
(2.5b)
with D0 the static or zero-frequency diffusion coefficient,
and c′ and c′′ positive constants.
Equation (2.4) was later derived theoretically by
Ernst, Hauge, and van Leeuwen,12 and by Dorfman and
Cohen.13 The basic physical idea behind the explanation
of the LTT phenomena is that it is the hydrodynamic
processes that determine the long-time behavior of all
correlation functions. Of particular importance are recollision processes, where after a collision, the two involved
particles diffuse away and then come back and recollide.
We can see this in Eq. (2.4), the right-hand side of which
is proportional to the probability that a diffusing particle
returns at time t to the point it started out from at t = 0.
This concept is a very general one, and it turns out that
all of the transport coefficients in a classical fluid have
LTT like those shown in Eqs. (2.4) and (2.5). That is,
if η(ω) is a general frequency dependent transport coefficient, then
η/ηB = 1 + a1,η n∗ + a′ 2,η (n∗ )2 log n∗ + a2,η (n∗ )2
+ o (n∗ )2
. (2.2)
Here o (n∗ )2 denotes a term that for n∗ → 0 vanishes
faster than (n∗ )2 . For real gases, only estimates of the
coefficient a′ 2,η are known.10 For the classical Lorentz
gas, a′ 2,D is known exactly.6
B. Long-time tails
Up until the late 1960’s it was also thought that the
equilibrium time correlation functions that determine the
η(ω)/η0 = 1 − c′η (iω)(d−2)/2 + . . .
2
(d > 2) , (2.6a)
η(ω)/η0 = 1 − c′′η log(iω) + . . .
(d = 2) .
last subsection, the spatial correlations decay exponentially in such systems, except at isolated critical points.
This asymmetry between space and time correlations is
related to the detailed balance relation that is valid in
equilibrium, and it is not generic. In more general states,
e.g. in nonequilibrium steady states, where detailed balance is absent and where there is spatial anisotropy, longrange correlations occur in both space and time.15
From a general point of view the existence of longranged spatial correlations is not surprising. Thermal
fluctuations constantly appear in a fluid, and then decay. Their behavior at long distances is determined by
the soft dynamical modes that are singular in the longwavelength limit. For example, in the static or zerofrequency limit, the diffusion equation becomes Laplace’s
equation whose solution exhibits a power-law decay in
space.
The best studied system, both theoretically and experimentally, is a simple fluid subject to a stationary temperature gradient.15,16 Using light scattering, the Fourier
transform of the density autocorrelation function,
(2.6b)
The existence of LTT even for arbitrarily dilute fluids
is not in contradiction with the Boltzmann equation becoming exact in the dilute limit. The point is that the
Boltzmann equation becomes exact for fixed time in the
limit of vanishing density, but not for fixed density, no
matter how small, in the limit of long times. The way
the dilute limit is reached is that with decreasing density,
one has to go to longer and longer times in order to see
the LTT, and the preasymptotic decay is well described
by the Boltzmann equation.
It was pointed out above that the LTT in a fluid are
related to the probability of a diffusing particle to return
at time t to the point where it was at time t = 0. Although the recollision events that are responsible for this
return probability always occur, this does not necessarily imply that all correlation functions in a given system
(other than a real classical fluid) decay as t−d/2 . Rather,
it just suggests the possibility that they do, and whether
or not a particular correlation function actually does so,
depends on whether the corresponding observable couples sufficiently strongly to the hydrodynamic processes
in the system under consideration. It is plausible to assume that, for any given observable, coupling to the diffusion process is more likely the more hydrodynamic or
soft modes there are in the system. For example, in a
d-dimensional real fluid there are d + 2 soft modes due to
the d + 2 conservation laws for particle number, energy,
and momentum. In contrast, in a Lorentz gas only particle number (and, trivially, energy) is conserved. This
smaller number of hydrodynamic modes suggests, according to the above argument, that the LTT might be weaker
in a Lorentz gas than in a real fluid. Indeed, the velocity
autocorrelation function in a Lorentz gas decays as,14
hv(0) · v(t)ieq ≈ −c (t0 /t)(d+2)/2
S(x, x′ ) = hδn(x) δn(x′ )i ,
can be measured. Here δn(x) is a density fluctuation at
point x, and the angular brackets denote a nonequilibrium ensemble average. Theoretically, both microscopic
many-body techniques and more phenomenological approaches yield,17


!2
k̂
·
∇T
c
⊥
p
 .
S(k) = S0 1 +
k2
T DT (ν + DT )
(2.10)
Here k is the wavevector, cp is the specific heat at constant pressure, DT is the thermal diffusion coefficient, ν
is the kinematic viscosity, k̂⊥ is a unit vector perpendicular to k, and S0 is the equilibrium static structure factor
that is k-independent in the long wavelength limit. Equation (2.10) for S(k) is well confirmed experimentally. For
the implications of Eq. (2.10) in real space, see Ref. 18.
Theoretically, long-range correlations in a nonequilibrium Lorentz gas have also been studied. The system
considered is a Lorentz gas in the presence of a chemical
potential gradient, ∇µ. As mentioned earlier, the only
soft mode in this model is the diffusive number density.
Because the diffusing particles in a Lorentz model do not
interact, correlations can only occur between the moving
particles and the scatterers. Denoting the density fluctuations of the former by δn(x), and the scatterer density
for a given configuration by N (x), a measure of this correlation is the particle-scatterer density correlation,
(for t >> t0 ) ,
(2.7)
and the frequency dependent diffusion coefficient is given
by
D(ω)/D0 = 1 + a iω + b (iω)d/2 + . . .
(d > 2) ,
(2.8a)
D(ω)/D0 = 1 + b′ iω log(iω) + . . .
(2.9)
(d = 2) . (2.8b)
The coefficients c, b, and b′ in Eqs. (2.7) and (2.8) are
positive. This sign occurs because in a Lorentz gas, in
contrast to a real fluid, any recollision process decreases
the diffusion rate. This is a consequence of the missing
dynamics of the scatterers.
S(x, x′ ) = hδn(x) N (x′ )i .
C. Long-range correlations in nonequilibrium steady
states
(2.11)
In Fourier space, a kinetic theory calculation yields,19
In contrast to the slow time decay of equilibrium time
correlation functions for classical fluids discussed in the
S(k) =
3
ik · ∇µ
2πDk2
,
(2.12)
+
+
+
1.0
+
+
with D the diffusion coefficient. In real space, Eq. (2.12)
implies that S(r) decays like r1−d . Again, the correlations are stronger in the real fluid than in the Lorentz
gas.
0.8
µ/µ cl
0.6
III. DISORDERED ELECTRONS AT ZERO
TEMPERATURE
+
+
0.4
+
+
A. Density expansion of the electron mobility
0.2
In disordered electronic systems, there are two length
scales that can be used to construct a dimensionless small
parameter from the scatterer density n, namely the scattering length σ and the Fermi wavelength λF = 2π/kF .
We can thus form the dimensionless densities (for d = 3)
nλ3F , nλ2F σ, nλF σ 2 , and nσ 3 . The first two do not appear
in the standard perturbation theory that expands in powers of n.8 The third one is usually written as 1/kF l ≡ ǫ,
with l ∼ 1/nσ 2 the mean-free path. The fourth one is
the quantity n∗ = nσ 3 that also has a classical meaning. Now consider a dilute electron system in the sense
that λF >> σ. In that case we have n∗ ∼ σ/l << ǫ,
and hence n∗ can be neglected compared to ǫ. Notice
that in the quantum Lorentz gas, the dilute limit needs
to be considered even for noninteracting quantum particles, since the Pauli principle establishes correlations,
and hence an effective interaction, between the particles.
For dimensions d > 2, the system is diffusive as long
as ǫ < ǫ∗ , with ǫ∗ = O(1) for d = 3. At ǫ = ǫ∗ a metalinsulator transition called the Anderson transition takes
place.20 For dilute systems in the above sense, n∗ << 1
even at the Anderson transition, and the latter is well
separated from the percolation transition that would occur at n∗ ≈ 1.
Considering the generalized virial expansion given by
Eq. (2.2) it is natural to ask whether or not a similar
expression holds for a quantum Lorentz gas with the parameter ǫ defined above replacing the classical reduced
density n∗ . This question was first asked, and answered
affirmatively, as early as 1966.21 More recently, all of
the terms in the expansion up to and including those
of O(ǫ2 ) have been computed exactly.22 In order to compare the resulting expression with real experimental data,
the electron mobility, µ, of electrons injected into Helium gas at low temperatures has been considered. Because the Helium atoms are very massive compared to
the electrons, the quantum Lorentz gas constitutes an
excellent model for this system. Furthermore, there is
experimental control over the density of the injected electrons, which means that the abovementioned diluteness
condition can be fulfilled to an extremely high degree.
This also means that Coulomb interaction effects can be
made negligible. In order to model the actual experimental situation, one needs to consider a nondegenerate gas
of electrons at temperature T with the thermal de Broglie
0.0
0.0
0.2
0.4
0.6
0.8
1.0
2/kTl
FIG. 1. Mobility µ of electrons in dense gases, normalized
to the Boltzmann value µcl , as a function of χ. The symbols
represent experimental data, the solid line is the theoretical
result, Eq. (3.1a). After Ref. 23.
wavelength, λT = 2π/kT = (2π/mkB T )1/2 , replacing λF
in ǫ. Defining χ = 2/kT l, an exact calculation gives,22
,
µ/µcl = 1 + µ1 χ + µ′2 χ2 log χ + µ2 χ2 + o χ2
(3.1a)
with
µ1 = −π 3/2 /6
,
= π 2 − 4 /32 ,
µ2 = 0.236 . . .
,
µ′2
(3.1b)
and µcl the classical or Boltzmann value for the mobility.
In Figs. 1 and 2 this theoretical result is compared with
experimental data. Of particular interest is the question
whether this kind of analysis can be used to experimentally confirm the existence of the logarithmic term in Eq.
(3.1a). In classical systems the corresponding logarithmic
term in Eq. (2.2) has never been convincingly observed,10
mostly due to the fact that the coefficients in the density expansion are not known exactly for any realistic
classical system. The fact that the quantum Lorentz gas
is such a good model for electrons in Helium gas makes
this system a very promising one for attempts to finally
observe the logarithm. As a measure of the logarithmic
term, one defines,22
f (χ) ≡ [ µ/µcl − 1 − µ1 χ ] /χ2
.
The theoretical prediction for this quantity is,
√
f (χ) = µ′2 log χ + µ2 ± µ2 2 πχ .
(3.2a)
(3.2b)
The last term in Eq. (3.2b) is an estimate of the effect
of all higher order terms. At T = 4.2K, a Helium gas
density of n = 1021 cm−3 corresponds to χ = 1, and
data were obtained for χ as small as 0.08. Fig. 2 shows
4
1
0.06
R/R - 1
f( χ )
0
0
–1
0.02
–2
–3
-0.02
–4
–5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-0.06
χ
0.40
0.63
1.00
1.58
2.51
T (K)
FIG. 2. The reduced mobility f , as defined in Eq. (3.2a),
vs. the density parameter χ = 2/kT l. The theoretical prediction is for f to lie between the two solid lines. The experimental data are from Fig. 9 of Ref. 24 with error bars estimated
as described in Ref. 22. The broken lines show what the theoretical prediction would be in the absence of the logarithmic
term in the density expansion. From Ref. 22.
FIG. 3. Resistance, R, normalized to R0 = R(T = 1K), of
a thin PdAu film plotted versus log T . After Ref. 28.
σ(t) ≈ −c (t0 /t)d/2
B. Long-time tails, a.k.a. weak localization effects
The results discussed in the previous subsection seems
to suggest that there is no conceptual difference between
transport in classical and dilute quantum Lorentz gases:
The forms of the density or disorder expansions, Eqs.
(2.2) and (3.1), are identical, even though the dimensionless expansion parameters are different. This conclusion
is fallacious, however, as can be seen by considering the
LTT in the time correlation functions, or in the low frequency expansions of the transport coefficients for the
two models. Here we first quote the quantum result, and
then we discuss the reason for it being qualitatively different from its classical counterpart.
Any of a variety of theoretical methods leads to a frequency dependent electrical conductivity for a quantum
Lorentz gas whose real part is of the form,25,26
σ(ω)/σ0 = 1 + c′ log ω + . . .
(for
t >> t0 ) .
d > 2) , (3.3a)
ψ(0)=−ψ(1/kB T )
(for
d = 2) .
(3.3c)
The coefficient c in Eq. (3.3c) is positive. For d = 2, and
more generally for d ≤ 2, the low frequency expansion
of σ(ω) breaks down, and the static conductivity or the
static diffusion coefficient is actually zero. That is, for
d ≤ 2 there is no metallic phase at zero temperature.27,20
At finite temperature and zero frequency, the temperature dependence of σ is obtained by replacing the frequency in Eqs. (3.3a) and (3.3b) by the temperature T .
For T > 0 and ω → 0 the leading frequency dependences are given by Eqs. (2.8), i.e., the classical result
is recovered. Figure 3 shows an example of the temperature dependent resistivity of a thin metallic film, which is
logarithmic for low temperatures in agreement with the
above remarks.
Let us now discuss the interesting difference between
the frequency dependencies in Eqs. (2.8) and (3.3). Equations (2.8) can be derived not only from a classical microscopic many-body approach, but also from a more general
phenomenological approach that appears to be independent of whether or not the underlying description is classical or quantum mechanical.29 The crucial assumption is
that the only slow mode in the problem is due to particle
number conservation. Remarkably, it is this assumption
that breaks down in the quantum case, and this is what
leads to the differences between Eqs. (2.8) and (3.3). To
understand this important point, let us consider a field
theoretic description of a disordered fermion system,30
that we assume to be noninteracting for simplicity. The
partition function is,
Z
Z=
D[ψ̄, ψ] exp S[ψ̄, ψ]
, (3.4a)
the theoretical prediction, Eq. (3.2b), for 0 < χ < 0.7
together with data by Schwarz. We conclude that the
existing experimental data are consistent with the theoretical result. However, for a convincing demonstration
of the existence of the logarithmic term an improvement
in the experimental accuracy by about a factor of ten
over Schwarz’s experiment would be necessary.
σ(ω)/σ0 = 1 + c ω (d−2)/2 + . . .
(for
(3.3b)
with the action S given by,
Z
∂
S[ψ̄, ψ] = − dx ψ̄(x)
ψ(x) +
∂τ
Equations (3.3a), (3.3b) imply that the current-current
correlation function that is defined as the Fourier transform of the real part of the conductivity has a LTT,
5
+
Z
dx ψ̄(x)
∆
+ µ − u(x) ψ(x)
2m
.
C. Long-ranged spatial correlations in equilibrium
(3.4b)
A characteristic feature of quantum statistical mechanics, as opposed to the classical theory, is the coupling of
statics and dynamics. This can be seen in Eqs. (3.4),
where the basic statistical field ψ is a function of both
space and imaginary time. As a result, one does not
expect any qualitative differences between static and dynamic correlations even in equilibrium, in contrast to the
asymmetry between these two types of correlations that
is observed in classical systems and was discussed in Sec.
II C above.
Indeed, a calculation of the wavenumber dependent
static spin susceptibility, χs (k), in a disordered system
of interacting electrons yields the following behavior for
small wavenumbers,33,34
Here we have used a four-vector notation, x ≡ (x, τ ),
R
R
R 1/T
dx ≡ dx 0 dτ , with τ denoting imaginary time.
m is the fermion mass, µ is the chemical potential, u(x)
is a random potential, and for notational simplicity we
have suppressed the spin labels. Since we are considering
fermions, the fields ψ̄ and ψ are Grassmann valued and
D[ψ̄, ψ] is a Grassmannian functional integration measure, but for the following arguments this will not be
crucial. By changing from imaginary time representation
to a frequency representation,
X
ψ(x) = T 1/2
e−iωn τ ψn (x) ,
(3.5a)
n
with Matsubara frequencies
ωn = 2πT (n + 1/2) ,
χs (k) = c0 − cd−2 |k|d−2 − c2 k2 + . . .
(n = 0, ±1, . . .) ,
(3.5b)
the action can be written,
XZ
1
∆ + µ − u(x) − iωn ψn (x)
S=
dx ψ̄n (x)
2m
n
.
The crucial point is that for ωn = 0, or T = 0, the
action given by Eq. (3.6) is invariant under a unitary
transformation
of the fields in frequency space, ψn →
P
U
ψ
.
In
fact, S is invariant under a larger symm nm m
plectic group that also includes a time reversal symmetry,
but we ignore this technical point here. We further note
that the ‘order parameter’,
ωn →0+
ωn →0−
(3.8)
where the ci are positive constants. In real space, the
nonanalytic term proportional to |k|d−2 , which for d < 4
is the leading k-dependence of χs , corresponds to a longrange interaction between the electronic spin density fluctuations that falls off like r2−2d . This has recently been
shown to have interesting consequences for the ferromagnetic quantum phase transition that occurs in an itinerant electron system at zero temperature as a function of
the exchange interaction.34
The origin of this long-range correlation can be traced
back to the same Goldstone modes that were discussed
in the last subsection, and that also lead to the LTT. In
a disordered system, the Goldstone modes are diffusive,
and give a contribution to the spin susceptibility that can
be schematically represented by
(3.6)
Q = lim hψ̄n (x) ψn (x)i − lim hψ̄n (x) ψn (x)i
,
,
χs (k) ∼
(3.7)
Z
k
Λ
dp p
d−1
Z
0
∞
dω
ω
(D p2 + ω)3
,
(3.9)
with Λ an ultraviolet cutoff, and D the spin diffusion coefficient. Equation (3.9) demonstrates the coupling of statics and dynamics that was mentioned above, and doing
the integrals yields Eq. (3.8). The fact that this coupling
is really a quantum effect can be seen by considering the
corresponding expression at finite temperature. In this
case one has to perform a frequency sum rather than an
integral, and the net effect is that the term |k|d−2 in Eq.
(3.8) is replaced by (k2 + T )(d−2)/2 . Hence, for T > 0 an
analytic expansion about k = 0 exists, and there are no
long-ranged correlations.
is the single-particle spectral function, or the difference
between the retarded and advanced Green functions. Because these functions have poles on opposite sides of the
real axis, Q is nonzero as long as the density of states at
the Fermi surface is nonzero.
To use a magnetic analogy, having a nonzero Q in Eq.
(3.7) is similar to having a nonvanishing magnetization
in the limit of a zero external field, and the abovementioned unitary symmetry is analogous to the rotational
symmetry in spin space. This analogy implies that in
the zero temperature fermion system there is a spontaneously broken continuous symmetry. This was first noticed in the context of the Anderson transition mentioned
in Sec. III A.31,32 Goldstone’s theorem then implies that
there are soft modes, namely particle-hole excitations, in
addition to the ones implied by the conservation laws.
Detailed calculations confirm that it is these additional
soft modes that lead to the stronger LTT effects in Eqs.
(3.3) as compared to the classical LTT in Eqs. (2.8).
D. Long-ranged spatial correlations in
nonequilibrium steady states
Very recently, spatial correlations of density fluctuations have been studied in noninteracting disordered electronic system that are not in equilibrium.35 For the model
6
defined by Eqs. (3.4) or (3.6) the correlation function
analogous to Eq. (2.11) for the classical Lorentz gas is,
S1 (x, x′ ) = {hδn(x)i u(x′ )}dis
,
one has to consider nonequilibrium states in order to get
long-ranged static correlations.
A phenomenon similar to the enhanced long-time tail
effects in the quantum case is also known in certain classical systems with soft modes that are unrelated to conservation laws. For example, in the smectic-A phase of liquid crystals there are soft modes due to the conservation
laws, and additional soft modes due to spontaneously
broken symmetries and Goldstone’s theorem. The combination of these soft modes produce stronger long-time
tail effects than are present in a simple classical fluid with
no Goldstone modes.36
Finally, we mention that the effects in electronic systems discussed in Sec. III do not qualitatively hinge on
the system being disordered. As can be seen from Eq.
(3.6) and the related discussion, the broken symmetry
argument still holds for clean electron fluids, with the
only difference being that the Goldstone modes have a
ballistic dispersion in that case, rather than a diffusive
one. Consequently, the effects discussed in this paper
qualitatively survive, only the various exponents change
compared to the disordered system. These LTT effects in
clean fermion systems can be related to known features
of Fermi-liquid theory. This demonstrates the generality
and unifying properties of the general physical approach
taken in this paper.
(3.10)
where {. . .}dis denotes the disorder average, and < . . . >
denotes a nonequilibrium thermal average as in Sec. II C.
A direct many-body calculation shows that the Fourier
transform of Eq. (3.10), S1 (k), behaves just like its classical counterpart, Eq. (2.12).
Because the moving particles are fermions, they effectively interact due to statistical correlations. As a measure of these correlations we consider the structure factor,
S2 (x, x′ ) = {hδn(x) δn(x′ )i}dis
.
(3.11)
Let us consider the nonequilibrium part of S2 . For a
classical, interacting, Lorentz gas one finds,
S2 (k) ∼ (∇µ)2 /k2
.
(3.12)
Naively, one might anticipate a similar result for the disordered electron system at T = 0. However, because of
the additional soft modes that were discussed in the preceding section, the correlations here are much stronger,
and the decay is much slower in space. A direct manybody calculation yields in the limit of small wavenumbers,
S2 (k) =
i
h
NF µτ
2
2
25(∇µ)
−
12(
k̂
·
∇µ)
6dπ(Dk2 )2
ACKNOWLEDGMENTS
,
It is our pleasure to dedicate this paper to Matthieu H.
Ernst on the occasion of his sixtieth birthday. Matthieu
has been one of the pioneers in discovering and understanding the physical phenomena discussed above. His
early work on classical systems laid much of the basis for
later developments, and he has remained at the forefront
of research in this field.
This work was supported by the National Science
Foundation under Grant Nos. DMR-92-17496 and DMR95-10185.
(3.13)
with Nf the electronic density of states at the Fermi level,
and τ the electronic mean-free time between collisions.
In real space, Eq. (3.13) corresponds to a linear decay of
S2 (r) with distance.
IV. CONCLUSION
In this paper we have reviewed classical and quantum
versions of what one might call generalized long-time tail
effects, that is long-range correlations in both space and
time. We have seen that these effects are due to the
hydrodynamic or soft modes in the system, which couple
via mode-mode coupling effects to all other modes unless
some symmetry prohibits such a coupling. Generally, the
quantum or zero temperature versions of these effects
are stronger, that is of longer range, than their classical
counterparts, because of additional soft modes that exist
at zero temperature. These additional soft modes are
Goldstone modes that result from a broken symmetry
in Matsubara frequency space, and are not related to
conservation laws. Another additional effect in quantum
systems is the coupling of statics and dynamics, which
leads to both static and dynamic equilibrium correlations
in general to be of long range, whereas in classical systems
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8